Theorem lubfun 17377

Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)

Hypothesis

Ref	Expression
lubfun.u	⊢ 𝑈 = (lub‘𝐾)

Assertion

Ref	Expression
lubfun	⊢ Fun 𝑈

Proof of Theorem lubfun

Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6175	. . . 4 ⊢ Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
2		funres 6179	. . . 4 ⊢ (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
3	1, 2	ax-mp 5	. . 3 ⊢ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})
4		eqid 2778	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2778	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6		lubfun.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
7		biid 253	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
8		id 22	. . . . 5 ⊢ (𝐾 ∈ V → 𝐾 ∈ V)
9	4, 5, 6, 7, 8	lubfval 17375	. . . 4 ⊢ (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
10	9	funeqd 6159	. . 3 ⊢ (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})))
11	3, 10	mpbiri 250	. 2 ⊢ (𝐾 ∈ V → Fun 𝑈)
12		fun0 6201	. . 3 ⊢ Fun ∅
13		fvprc 6441	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (lub‘𝐾) = ∅)
14	6, 13	syl5eq 2826	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝑈 = ∅)
15	14	funeqd 6159	. . 3 ⊢ (¬ 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
16	12, 15	mpbiri 250	. 2 ⊢ (¬ 𝐾 ∈ V → Fun 𝑈)
17	11, 16	pm2.61i 177	1 ⊢ Fun 𝑈

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {cab 2763  ∀wral 3090  ∃!wreu 3092  Vcvv 3398  ∅c0 4141  𝒫 cpw 4379   class class class wbr 4888   ↦ cmpt 4967   ↾ cres 5359  Fun wfun 6131  ‘cfv 6137  ℩crio 6884  Basecbs 16266  lecple 16356  lubclub 17339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-lub 17371

This theorem is referenced by:  joinfval  17398  joinfval2  17399